to mathematically characterize the population. In this course, we’ll assume
that our sample is a random draw from the population.
So our definition is that a random variable is a numerical outcome of an
experiment. The random variables that we study will come in two varieties,
discrete or continuous. Discrete random variables are random variables
that take on only a countable number of possibilities. Mass functions will
assign probabilities that they take specific values. Continuous random
variable can conceptually take any value on the real line or some subset of
the real line and we talk about the probability that they lie within some
range. Densities will characterize these probabilities.
Let’s consider some examples of measurements that could be considered
random variables. First, familiar gambling experiments like the tossing of a
coin and the rolling of a die produce random variables. For the coin, we
typically code a tail as a 0 and a head as a 1. (For the die, the number facing
up would be the random variable.) We will use these examples a lot to help
us build intuition. However, they aren’t interesting in the sense of seeming
very contrived. Nonetheless, the coin example is particularly useful since
many of the experiments we consider will be modeled as if tossing a biased
coin. Modeling any binary characteristic from a random sample of a
population can be thought of as a coin toss, with the random sampling
performing the roll of the toss and the population percentage of individuals
with the characteristic is the probability of a head. Consider, for example,
logging whether or not subjects were hypertensive in a random sample.
Each subject’s outcome can be modeled as a coin toss. In a similar sense the
die roll serves as our model for phenomena with more than one level, such
as hair color or rating scales.
Consider also the random variable of the number of web hits for a site each
day. This variable is a count, but is largely unbounded (or at least we
couldn’t put a specific reasonable upper limit). Random variables like this
are often modeled with the so called Poisson distribution.
Finally, consider some continuous random variables. Think of things like
lengths or weights. It is mathematically convenient to model these as if they
were continuous (even if measurements were truncated liberally). In fact,